---- > [!theorem] Theorem. ([[the convolution theorem for Fourier series]]) > For [[Riemann integral|Riemann integrable]] $f$ [[function on the (unit) circle|on the circle]], we have that $(\widehat{f * g})(n)=\hat{f}(n)\hat{g}(n),$ > where $f * g$ denotes the [[convolution]] of $f$ with $g$. ^c2ec48 > [!proof]- Proof. ([[the convolution theorem for Fourier series]]) > $\begin{align} (\widehat{\textcolor{Skyblue}{f * g}})(n)= & \int _{t=0} ^{t=1} \left[\textcolor{Skyblue}{\int_{s=0}^{s=1} f(s)g(t-s)\, \, ds \,} \right] \textcolor{Apricot}{e ^{-2\pi i n t}} \, dt \\ = & \textcolor{Skyblue}{\int_{s=0} ^{s=1}} \int_{t=0} ^{t=1} \textcolor{Skyblue}{f(s)}\textcolor{Apricot}{e ^{-2\pi i n s}}\textcolor{Skyblue}{g(t-s)} \textcolor{Apricot}{e ^{-2\pi i n (t-s)}} \, \, dt \ \textcolor{Skyblue}{ds} \text{ (Fubini)}\\ = & \int _{s=0} ^{s=1} f(s) e ^{-2\pi i n s} \underbrace{\left[\int_{t=0}^{t=1} g(t-s) e ^{-2\pi i n (t-s)}\, dt \right]}_{{}=\int _{0} ^{1} g(t) e ^{-2\pi i n t} \, dt, \text{ by periodicity} } \, ds \\ = & \left( \int_{s=0}^{s=1} f(s) e ^{-2\pi i n s}\,ds \right)\left( \int_{t=0}^{t=1}g(t) e ^{-2\pi i n t}\, dt \right) \\ =& \hat{f}(n) \hat{g}(n), \end{align}$ where we apply [[Fubini's Theorem]] to swap [[Riemann integral|integrals]]. ^33afb5 ---- #### ----- #### References > [!backlink] > ```dataview TABLE rows.file.link as "Further Reading" FROM [[]] FLATTEN file.tags GROUP BY file.tags as Tag > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```